The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups. (English) Zbl 1498.43005
Summary: We show that the \(\beta \)-numbers of intrinsic Lipschitz graphs of Heisenberg groups \(\mathbb{H}_n\) are locally Carleson integrable when \(n\geq 2\). Our main bound uses a novel slicing argument to decompose intrinsic Lipschitz graphs into graphs of Lipschitz functions. A key ingredient in our proof is a Euclidean inequality that bounds the \(\beta \)-numbers of the original graph in terms of the \(\beta \)-numbers of many families of slices. This allows us to use recent work of K. Fässler and T. Orponen [Bull. Lond. Math. Soc. 52, No. 3, 472–488 (2020; Zbl 1457.46046)] which asserts that Lipschitz functions satisfy a Dorronsoro inequality.
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 28A75 | Length, area, volume, other geometric measure theory |
| 05C99 | Graph theory |
| 53C17 | Sub-Riemannian geometry |
Citations:
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